I found myself pondering about space and realized that aside from the general notion of dimension (that being the minimum number of coordinates needed to specify an entity in the space under consideration), I can only think of dimensions greater than four in an algebraic manner, but not in any other way. From my conversations with others, along with plain intuition, it seems that this is the case for all humans (or at least everyone I had a chance to speak with regarding this).

The Cartesian system was a byproduct of the realization that regular Euclidean space can be "arithmatized" if you will. Yet it was ultimately representing an abstraction of "space" as can be imagined (as regular Euclidian space was merely an idealization of space as we know it) and could be represented (most often) in synthetic form. Yet it seems when considering multiple dimensions beyond what the human mind can conjure visually, it must be represented in a fully algebraic/arithmatical manner.

I find this to be quite interesting if not disturbing, as ostensibly once you enter the realm of raw analytic geometry such that it is impossible to represent the notion synthetically, you are no longer speaking about "space" in its initial primitive sense (the notion of space that was initially abstracted), but rather speaking about a mathematical formalism that has a (very) loose connection with space as initially understood.

Hence the question is, are questions regarding the possibility or impossibility of "n-dimensional" spaces (being instantiated in reality) simply missing the mark? That is to say, are such questions equivocating between, what seems to be, a language "game" and visual/physical space? It seems almost akin to asking about the possibility of surreal number arithmetic being physically instantiated.

Perhaps I'm missing something crucial, any thoughts would be appreciated.

-Thank you

  • Cartesian system goes back to ancient geometers like Apollonius, and can be set up geometrically using families of lines, without arithmetization. And while we may not be able to picture 4D shapes mentally (we cannot even picture infinite 2D lines), modern geometers are quite adept at reasoning about them synthetically, see surgery theory, for example. As such reasoning perfectly (and provably) agrees with coordinate calculations, calling it "formalism that has a very loose connection with space as initially understood" hardly seems fair.
    – Conifold
    Commented May 10 at 5:42
  • @Conifold As I understand it, the defining property of Cartesian systems is arithmatization. Regarding the latter portion, I suspect you are falling into the pit that I was attempting to shine light on. The reasoning that's occurring regarding these objects is purely done in an arithmetical/algebraic/set-theoretic manner; this much is undeniable (I don't see how surgery theory provides an exception to this). The point was that it seems odd to speak of (what seems to be) purely formalistic notions as such being instantiated in reality; such questions seem misplaced an incomprehensible to me.
    – Max Maxman
    Commented May 10 at 17:03
  • @Conifold Following up, it seems that my usage of analytic and synthetic is vague. Read synthetic as "linked with space perception" and analytic as "not linked with space perception".
    – Max Maxman
    Commented May 10 at 17:55
  • 1
    As I see it, arithmetical and set-theoretic are different things. Synthetic reasoning in 2D Euclidean geometry is also set-theoretic these days. And there is clear continuity between it and diagrammatic reasoning in hyperbolic geometry by Bolyai and Lobachevsky, and with reasoning over modern surgery diagrams. According to psychologists, visual space is not Euclidean either, see Hatfield. Euclidean geometry is produced by intuitive idealization, and that can produce more. So I am inclined to deny the "purely arithmetical/algebraic manner".
    – Conifold
    Commented May 11 at 5:16

3 Answers 3


It is indeed a bit of a trip to find out that our intuitions about space being three-dimensional might simply arise from the linear basis of the two 2-dimensional array of nerves that form the transduction into our visual system and are not inherent properties of some physically real thing itself. It's the same way when you learn that a tree falling in the forest may disturb the air with waves, but does not actually produce a sound which is done by the hearer's auditory system. For thousands of years, Euclid's geometry was seen as a description of space itself, and both the Newtonian notion of absolute space and time, seemed to be real definitions, rather than near-universal conventions to be superseded by more adequate models of space and time that Einstein's theory of relativity provided.

You ask:

are questions regarding the possibility or impossibility of "n-dimensional" spaces (being instantiated in reality) simply missing the mark? That is to say, are such questions equivocating between, what seems to be, a language "game" and visual/physical space?

What you are noticing is the contention between realist and instrumentalist metaphysics. For a realist, space has an essential trait of 3-dimensions, but for an instrumentalist, 3-dimensions is just a useful description that carries with it no ontological commitment. From WP:

In philosophy of science and in epistemology, instrumentalism is a methodological view that ideas are useful instruments, and that the worth of an idea is based on how effective it is in explaining and predicting natural phenomena. According to instrumentalists, a successful scientific theory reveals nothing known either true or false about nature's unobservable objects, properties or processes.1 Scientific theory is merely a tool whereby humans predict observations in a particular domain of nature by formulating laws, which state or summarize regularities, while theories themselves do not reveal supposedly hidden aspects of nature that somehow explain these laws.

This is an important philosophical distinction, and the roots of this idea in more modern philosophy can be ascribed partially to the ontological ambiguity that Kantian philosophy provides in the idea of the Ding an sich; roughly, it suggests that our knowledge of things, should they be real and independent of us, are always through the lens in which we view them. Should we maintain they are real and independent of us, then we accept scientific realism. Thus, we may loosen our claims that space is 'real' and has essential properties, and may rather accept that 'space' has a description which may or may not be adequate depending on our goals.

If you're curious, I believe the last survey I read by PhilPapers cited that approximately 75% of respondents considered themselves realists in this regard. So, does the question miss the mark? I would say it depends on who you ask. There are some contemporary philosophical positions, like contemporary conceptualism, that argue that not only are these instrumentalist theories influential, that they literally shape our perceptual apparatus. If that's the case, then the very notion there are various conceptual theories that have varying degrees of adequacy impact our very intuitions as we accept them.


That we can’t clearly visualize in our mind’s eyes n-dimensional space for n > 3 doesn’t imply that our understanding of it must be purely algebraic. Indeed, that strikes me as one of the themes of Abbott’s Flatland.

We can visualize for instance projections of higher-dimensional objects onto 3-space, and one can develop a pretty respectable intuition by experimenting with rotating, say a tesseract (i.e., the 4-D analog of a cube) about various planes. There are lots of animations on line that illustrate this; two of them can be found in the Wikipedia article on tesseracts.


Dimensions of physical OBJECTS, and dimensions of SPACE are not the same thing. It seems you are just focusing on 3D physical objects. Objects are empirical, time and space are ideals. You can't perceive n-dimensional space (including 3D space) as such, unless you put a thing of dimension n within.

The dimensions of physical objects are always three, no more and, believe it or not, no less. There are no physical objects of dimensions 2 or 1. A line or a triangle are not physical objects. A country or a frontier are just ideals. When you cut an apple and believe looking at a flat surface, news update: the surface is not flat at all.

But the dimensions of ideal/rational/metaphysical objects are not always three. A line has dimension 1, a dot has dimension 0, a circle has dimension 2 and a cube has dimension 3. Space has three dimensions only because we know bodies of three dimensions.

But it is easy to perceive more ideal/rational/metaphysical dimensions. For example, time can be a fourth dimension. Move your hand to the left and then, to the right. Now, in your imagination, repeat the movement several times. There you have. You're looking at your hand in four dimensions. You want more? You've added time, and now, add color. Imagine the same progression, but now, include coloring: you can travel the time progression forwards and backwards (time dimension) in different colors (color dimension). There you have: your hand in five dimensions.

Now, you will say, yes, but I mean physical dimensions. I want to perceive, physically, my car in four dimensions.

Sorry, you can't. That's simply because you, the subject, exist physically also as a three-dimensional object.

Perceiving implies an interaction, and interactions occur ALWAYS between a subject (you) and an object (your car). If the subject won't exist, the dimensions of the object would not be limited. For example, your car would be infinitely small or large, infinitely dense or not. It all depends on what would be the dimensions of a new observer (subject). If you are created as an observer in an infinite 2d plane which intersects with the object, you will perceive a flat form. If you would be created as a subject which can change its density, you would be able to enter the car without opening the door, and at the same time, your finger could completely explode if you touch the car, like what happens with water that enters in contact with a wall.

Now, you understand that £the subject determines the object_. There could be eleven dimensions where you really exist, but you just can't perceive them because of your own attributes. Physics tell us of such dimensions, and they are not only ideal, but real.

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